Explicit construction of entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadratic flow-density relationship

Transportation Research Part B: Methodological

Volumn 42, Issue 4, 2008, Pages 355-372

  Lu, Yadong ^a   Wong, S C ^b   Zhang, Mengping ^a   Shu, Chi Wang ^c   Chen, Wenqin ^b  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b UNIVERSITY OF HONG KONG   (Hong Kong)

^c BROWN UNIVERSITY   (United States)

Author keywords

Entropy solution; LWR model; Piecewise quadratic flow density relationship; Traffic flow; WENO scheme

Indexed keywords

BOUNDARY CONDITIONS; MATHEMATICAL MODELS; PIECEWISE LINEAR TECHNIQUES; POLYNOMIALS;

ENTROPY SOLUTION; PIECEWISE QUADRATIC FLOW-DENSITY RELATIONSHIP; TRAFFIC FLOW;

TRAFFIC CONTROL;

BOUNDARY CONDITION; NUMERICAL MODEL; TESTING METHOD; TRAFFIC MANAGEMENT; TRANSPORTATION PLANNING;

EID: 39749105993     PISSN: 01912615     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.trb.2007.08.004     Document Type: Article

References (36)

1. M. Abramowitz I.A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972 Dover Publications New York
2. R. Ansorge What does the entropy condition mean in traffic flow theory Transportation Research Part B 24 2 1990 133 143
3. C.M. Dafermos Polygonal approximations of solutions of initial value-problem for a conservation law Journal of Mathematical Analysis and Applications 38 1 1972 33 41
4. C.F. Daganzo The cell transmission model – a dynamic representation of highway traffic consistent with the hydrodynamic theory Transportation Research Part B 28 4 1994 269 287
5. C.F. Daganzo A finite difference approximation of the kinematic wave model of traffic flow Transportation Research Part B 29 4 1995 261 276
6. C.F. Daganzo Fundamentals of Transportation and Traffic Operations 1997 Elsevier Science Ltd. Oxford, UK
7. C.F. Daganzo A variational formulation of kinematic waves: basic theory and complex boundary conditions Transportation Research Part B 39 2 2005 187 196
8. C.F. Daganzo A variational formulation of kinematic waves: solution methods Transportation Research Part B 39 10 2005 934 950
9. A.C. Dick Speed/flow relationships within an urban area Traffic Engineering and Control 8 6 1966 393 396
10. L.C. Edie Car-following and steady-state theory for noncongested traffic Operations Research 9 1 1961 66 76
11. B.D. Greenshields A study of traffic capacity Proceedings of Highway Research Board 14 1934 448 477
12. R. Haberman Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow 1977 Prentice-Hall Inc. Englewood Cliffs, New Jersey
13. H. Holden N.H. Risebro Front Tracking for Hyperbolic Conservation Laws 2002 Springer New York
14. G.S. Jiang C.W. Shu Efficient implementation of weighted ENO schemes Journal of Computational Physics 126 1 1996 202 228
15. Lax, P., 1973. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Regional Conference Series in Applied Mathematics, # 11. SIAM, Philadelphia.
16. J.P. Lebacque The Godunov scheme and what it means for first order traffic flow models J.B. Lesort Proceedings of the 13th International Symposium on Transportation and Traffic Theory 1996 Elsevier Science Ltd. Lyon, France 647 677
17. R.J. LeVeque Numerical Methods for Conservation Laws Lecture in Mathematics ETH Zurich 1992 Birkhauser Verlag Basel, Switzerland
18. M.J. Lighthill G.B. Whitham On kinetic wave II: a theory of traffic flow on crowded roads Proceedings of the Royal Society of London, Series A 229 1178 1955 317 345
19. B.J. Lucier A moving mesh numerical method for hyperbolic conservation laws Mathematics of Computation 46 1986 59 69
20. X.-D. Liu S. Osher T. Chan Weighted essentially nonoscillatory schemes Journal of Computational Physics 115 1 1994 200 212
21. H. Lo A cell-based traffic control formulation: strategies and benefits of dynamic timing plans Transportation Science 35 2 2001 148 164
22. H. Lo Y.W. Szeto A cell-based variational inequality formulation of the dynamic user optimal assignment problem Transportation Research Part B 36 5 2002 421 443
23. A.D. May H.E.M. Keller Evaluation of single and multi-regime traffic flow models W. Leutzbach P. Baron Proceedings of the Fourth International Symposium on Theory of Traffic Flow 1968 Karlsruhe Germany
24. P.G. Michalopoulos D.E. Beskos J.K. Lin Analysis of interrupted traffic flow by finite-difference methods Transportation Research Part B 18 4-5 1984 409 421
25. G.F. Newell A simplified theory of kinematic waves in highway traffic, 1. general-theory Transportation Research Part B 27 4 1993 281 287
26. G.F. Newell A simplified theory of kinematic waves in highway traffic, 2. queueing at freeway bottlenecks Transportation Research Part B 27 4 1993 289 303
27. G.F. Newell A simplified theory of kinematic waves in highway traffic, 3. multi-destination flows Transportation Research Part B 27 4 1993 305 313
28. P.I. Richards Shock waves on the highway Operations Research 4 1 1956 42 51
29. C.-W. Shu High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD International Journal of Computational Fluid Dynamics 17 2 2003 107 118
30. C.-W. Shu S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes Journal of Computational Physics 77 2 1988 439 471
31. R.T. Underwood Speed, volume and density relationships Quality and Theory of Traffic Flow, Bureau of Highway Traffic 1961 Yale University New Haven, CT
32. S. Velan M. Florian A note on the entropy solutions of the hydrodynamic model of traffic flow Transportation Science 36 4 2002 435 446
33. G.B. Whitham Linear and Nonlinear Waves 1974 John Wiley & Sons New York, USA
34. G.C.K. Wong S.C. Wong A multi-class traffic flow model – an extension of LWR model with heterogeneous drivers Transportation Research Part A 36 9 2002 827 841
35. S.C. Wong G.C.K. Wong An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed-density relationship Transportation Research Part B 36 8 2002 683 706
36. M. Zhang C.W. Shu G.C.K. Wong S.C. Wong A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model Journal of Computational Physics 191 2 2003 639 659